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| '''Wackersalish''' is the proto-language for !Zoom. | | '''Elements of Harmony''' ([[Windermere/Classical|Classical Windermere]]: ''Yămyămał clisăyfäl'') is a textbook on just intonation authored in [[Windermere/Classical|Classical Windermere]] by physicist, mathematician and composer Tsăhongtamdi covering elementary number theory, acoustics, and just intonation music theory. |
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| {| class="wikitable"
| | Supporters of [[Verse:Tricin/Plud Schrog-Hahn|Soha Plu]] believe that the text was written in [[Schlaub]] before it was translated into Classical Windermere, and that Tsăhongtamdi is simply a Windermere pseudonym of a (probably) Hlou composer. |
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| ! colspan="2" rowspan="2" |
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| ! rowspan="2" | [[Bilabial consonant|Bilabial]]
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| ! colspan="2" | [[Alveolar consonant|Alveolar]]
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| ! rowspan="2" | [[Postalveolar consonant|Post-<br>alveolar]]
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| ! rowspan="2" | [[Palatal consonant|Palatal]]
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| ! colspan="2" | [[Velar consonant|Velar]]
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| ! colspan="2" | [[Uvular consonant|Uvular]]
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| ! rowspan="2" | [[Glottal consonant|Glottal]]
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| |- style="font-size: x-small"
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| ! [[Central consonant|central]]
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| ! [[Lateral consonant|lateral]]
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| ! plain
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| ! [[Labialisation|labial]]
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| ! plain
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| ! [[Labialisation|labial]]
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| |-class=IPA align=center
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| ! rowspan="2" | [[Nasal consonant|Nasal]]
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| ! style="font-size: x-small;"| plain
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| | m
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| | n
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| | ŋ /ɴ/
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| |-class=IPA align=center
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| ! style="font-size: x-small;" | [[glottalic consonant|glottalized]]
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| | mʼ
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| | nʼ
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| | ŋ /ɴʼ/
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| |-class=IPA align=center
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| ! rowspan="2" | [[Plosive consonant|Plosive]]
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| ! style="font-size: x-small;" | plain
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| | p
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| | t
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| | (k)
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| | kʷ
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| | q
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| | qʷ
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| |-class=IPA align=center
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| ! style="font-size: x-small;" | glottalized
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| | pʼ
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| | tʼ
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| | kʼʷ
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| | qʼ
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| | qʼʷ
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| | ʔ
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| |-class=IPA align=center
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| ! rowspan="2" | [[Affricate consonant|Affricate]]
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| ! style="font-size: x-small;" | plain
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| | c /t͡s/
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| | č /t͡ʃ/
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| |-class=IPA align=center
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| ! style="font-size: x-small;" | glottalized
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| | cʼ /t͡sʼ/
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| | ƛʼ /t͡ɬʼ/
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| | čʼ /t͡ʃʼ/
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| |-class=IPA align=center
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| ! colspan="2"|[[Fricative consonant|Fricative]]
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| | s
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| | ɬ
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| | š /ʃ/
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| | xʷ
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| | x̣ /χ/
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| | x̣ʷ /χʷ/
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| | h
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| |-class=IPA align=center
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| ! rowspan="2" | [[Approximant consonant|Approximant]]
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| ! style="font-size: x-small;"| plain
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| | l
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| | y /j/
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| | w
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| |-class=IPA align=center
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| ! style="font-size: x-small;"| glottalized
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| | yʼ /jʼ/
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| | wʼ
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| |}
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| {| class="wikitable" style=text-align:center
| | ==Contents== |
| |-style="font-size: 90%;"
| | *Book 1 discusses mathematical results: |
| !colspan=2 rowspan=2|
| | **Prime factorization |
| !rowspan=2| [[Bilabial consonant|Bilabial]]
| | **Continued fractions and mediants |
| !colspan=2| [[Alveolar consonant|Alveolar]]
| | *Book 2 discusses basic acoustics (don't mention frequencies) |
| !rowspan=2| [[Palatal consonant|Palatal]]
| | **monochord; building it |
| !colspan=2| [[Velar consonant|Velar]]
| | **Mersenne's Laws? |
| !colspan=2| [[Uvular consonant|Uvular]]
| | **harmonic series |
| !rowspan=2| [[Glottal consonant|Glottal]]
| | **intervals as rational string length ratios (given equal thickness and tension); these can be written as tuples by unique factorization |
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| | *Book 3 discusses just intonation scales built from notes taken from overtone and undertone series. |
| ! <small>[[Central consonant|central]]</small>
| | **odd- and prime-limit |
| ! <small>[[Lateral consonant|lateral]]</small>
| | **chord voicings |
| ! <small>plain</small>
| | **The tonality diamond is described as a way to "connect" overtone chords/scales over different fundamentals. |
| ! <small>[[Labialization|labial]]</small>
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| ! <small>plain</small>
| | Tsăhongtamdi's most influential recommendation was against using fixed-pitch instruments; he argued that they were expressively limited. This recommendation was lasting in influence - most instruments used in traditional Hlou-Shum music are flexible-pitch instruments. |
| ! <small>labial</small>
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| |-
| | ==Full text (Schlaub)== |
| !rowspan=2| [[Nasal consonant|Nasal]]
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| ! <small>plain</small>
| | ==Full text (Classical Windermere)== |
| | {{IPAlink|m}}
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| | {{IPAlink|n}}
| | ==Full text (English)== |
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| | ==Sketch== |
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| | <!-- |
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| | '''Motivation for log''': Assume strings 1 and 2 have equal tension and thickness, and have lengths $L_1$ and $L_2$ respectively. Find a function $\log$ that, given the ratio between $L_1$ and $L_2$, measures the corresponding subjective difference in pitch. |
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| | $\log(\frac{L_1}/{L_2})$ gives the subjective intervallic change when moving from string 1 to string 2. |
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| | Thus: if $r_1$ and $r_2$ are string length ratios, then we want |
| ! <small>[[Glottalization|glottal.]]</small>
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| | mˀ
| | $\log(1) = 0;$ $\log(a) > 0$ when $a > 1;$ $\log(r_1 r_2) = \log(r_1) + \log(r_2).$ |
| | nˀ
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| | The area $A(t)$ under the hyperbola $y = \frac{1}/{x}$ from $x = 1$ to $x = t$ satisfies the desired properties: $A(1) = 0; A(tu) > A(t) + A(u)$. (Give a geometric proof) |
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| | Ts-T derives using calculus that the desired function is well-approximated by Taylor polynomials: namely, $log(x+1) = A(x+1) ≈ \sum_{n=1}^{N} (-1)^n \frac{x^n}/{n}$, when $|x| < 1$. |
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| | --> |
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| | [[Category:Tricin]] |
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| !rowspan=2| [[Stop consonant|Stop]]
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| ! <small>plain</small>
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| | {{IPAlink|p}}
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| | {{IPAlink|t}}
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| | {{IPAlink|tʃ}}
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| | {{IPAlink|k}}
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| | kʷ
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| | {{IPAlink|q}}
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| | qʷ
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| | {{IPAlink|ʔ}}
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| |-
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| ! <small>[[Ejective consonant|ejec.]]</small>
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| | {{IPAlink|pʼ}}
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| | {{IPAlink|tʼ}}
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| | {{IPAlink|tʃʼ}}
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| | {{IPAlink|kʼ}}
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| | kʷʼ
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| | {{IPAlink|qʼ}}
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| | qʷʼ
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| |-
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| !colspan=2| [[Fricative consonant|Fricative]]
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| | {{IPAlink|s}}
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| | {{IPAlink|ɬ}}
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| | {{IPAlink|x}}
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| | xʷ
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| | {{IPAlink|χ}}
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| | χʷ | |
| | {{IPAlink|h}} | |
| |-
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| !rowspan=2| [[Approximant consonant|Approxi-<br>mant]]
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| ! <small>plain</small>
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| | {{IPAlink|l}}
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| | {{IPAlink|j}}
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| | {{IPAlink|w}}
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| |-
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| ! <small>glottal.</small>
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| | lˀ
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| | jˀ
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| | wˀ
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| |}
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